Symbolic Logic
intellects
phi103_chapter06.pdf
6
Elementary Symbolic Logic
Heinz-Dieter Falkenstein
Against logic there is no armor like ignorance.
—Laurence Peter
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CHAPTER 6Section 6.1 The Logic of Sentences
Now that we have looked at inductive arguments, deductive arguments, and the com-ponents—premises and conclusions—that make them up, we can turn to making our understanding of deductive reasoning more precise. We do so by introducing a few sym- bols for sentences and for the kinds of terms—such as and and or—that connect sentences, and we use these symbols to look at the structure of both sentences and arguments. For the most part, we look at material we have already examined; here we just use symbols to make the various structures involved a bit more explicit. Arguments often get sidetracked because of the information presented, and although that information is important, we can avoid being sidetracked in this way by focusing on the structure of the arguments. Sym- bols are helpful in keeping the focus on such structures.
What We Will Be Exploring
• We will look at how symbols can be applied to sentences, and then to arguments. • We will examine the notion of a truth function and the kinds of specific logical properties sentences
possess. • We will see how truth tables can be used to evaluate certain kinds of sentences, as well as testing
deductive arguments for validity. • We will use basic symbolic logic to examine some earlier material and see how it can be made more
precise.
6.1 The Logic of Sentences
We begin by seeing how to apply symbols—sentence letters—to sentences, first using basic sentences and then using sentences constructed out of these basic sentences.
Assertoric Sentences and Sentence Letters
Earlier we looked at various strings of words: some were questions, some were commands, and some were assertions. Here are some examples of the kinds of sentences we examined earlier, as well some new, compound sentences, which assert more than one claim:
1. Cheddar cheese is better than American cheese. 2. The window is broken. 3. Turn left at the next light. 4. Art is a great dancer. 5. Art is very popular. 6. Art is very popular and he is a great dancer. 7. Art is very popular because he is a great dancer. 8. John and Mary got married. 9. John and Mary had a baby. 10. John and Mary got married and had a baby. 11. John and Mary had a baby and got married. 12. Can you hear that music?
Can you determine which of these make assertions or state some kind of claim? You’ll find that most of the sentences on this list do. Sentence 3, however, is an imperative, and
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CHAPTER 6Section 6.1 The Logic of Sentences
sentence 12 is a question; we are not able to evalu- ate those sentences, because they do not put forth claims that can be evaluated as true or false; that is, they are not assertoric sentences. In the type of logic we discuss in this text, we are able to evalu- ate only assertoric sentences. (Although we do not look at them, other logics have been devel- oped to consider such things as imperatives and questions.)
A few of these sentences, however, need spe- cial mention. Consider the sentences about Art, his dancing skills, and his popularity. Sentences 4 and 5 are both assertoric sentences; they each make claims that can be treated as either true or false. When these sentences are put together in sentence 6, we have a compound sentence that asserts two claims, but because of the way they are put together—with “and” connecting them—we can continue to treat sentence 6 as an assertoric sentence. But what about sentence 7?
Let’s imagine that we know it is, in fact, true that Art is a great dancer and that Art is very popular. If so, we know that the compound sentence—Art is a great dancer and he is very popular—is also true. But do we know that the reason Art is popu- lar is his skill at dancing? Could it be the case that he is a fabulous dancer, very popular, and yet be popular for some other reason than his dancing ability? That seems quite pos- sible. So what we see is that we may know that both component sentences—sentences 4 and 5—are true, but still not know that sentence 7 is true. Again, this is due to the way the sentences are put together. In sentence 7, they are put together using “because”; since we don’t have enough information to know that the reason Art is popular is his dancing skill, we can’t determine the truth of the sentence as a whole. We see some more examples of this a bit later; at this point, however, we introduce the idea of a truth function. The idea behind a truth function is simply that if we know the truth-values of the components of a sentence, the truth-value of complex sentences put together out of these components can be determined. For instance, if I know it is true that it is raining, and it is true that there is thunder, I also know it is true that it is raining and that there is thunder. For the reason we just saw, we cannot treat sentence 7 truth functionally: we don’t know the truth-value of the entire sentence simply by knowing the truth-values of its components, due to the term “because” that connects those two component sen- tences. In contrast, sentence 6 can be treated truth functionally: we can determine its truth-value by knowing the truth-values of its components, due to the term “and” that connects the two component sentences. All of this is to make what is really one simple point: “and” can be treated in a way distinct from “because,” and this difference means that “and” can be treated as a truth-functional connective and “because” cannot.
Thinkstock
Logicians try to determine whether they can treat a sentence truth func- tionally by determining its truth-value.
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CHAPTER 6Section 6.1 The Logic of Sentences
Highlights: What Is a Truth Function?
Let’s assume it is true that Kirk likes hot dogs and that it is also true that Kirk likes hamburgers. If so, we can determine that the more complex sentence Kirk likes hot dogs and hamburgers is true.
If we can determine the truth-value of a complex sentence just by knowing the truth-value of the components that make up the complex sentence, we can say that the complex sentence can be treated truth functionally. A truth function is a function that permits one to go from a set of truth-values to another truth-value. But it is probably easier to think of it this way: a sentence can be treated truth functionally if by knowing the truth-value of the components of a longer sentence, we can determine the truth-value of the longer sentence.
For instance, let’s assume Sacramento is the capital of California; so it is true that Sacramento is the capital of California. But perhaps Paul isn’t very good at geography and believes that Los Angeles is the capital of California. Using sentence letters, let “S” be “Sacramento is the capital of California,” and “L” be “Los Angeles is the capital of California.” If we know that “S” is true, and that “L” is false, we are not able to determine the truth-value of these sentences:
Paul believes that L.
Paul believes that S.
Since we cannot determine the truth-value of the longer sentence by knowing the truth- value of its component sentence and the phrase “Paul believes that,” we can’t go from the truth-value of the component sentences to the truth-value of the longer sentence. So this isn’t a truth function, and this also shows (for some complicated reasons) that “Paul believes that” cannot be treated truth functionally.
If we were to run into a sentence such as “Paul believes that L,” we would just symbolize it as “P” and treat it as an entire sentence.
Logicians typically use the letters “P” and “Q” to represent sentences (if we need more letters, we just continue along the alphabet, “R,” “S,” etc.). We can represent any assertoric sentence—which we call simply “sentences” from now on—with a sentence letter, such as “P.” So “Cheddar cheese is better than American cheese” can be represented as “P.” Simi- larly, here are some of the other sentences from our list represented with sentence letters:
1. Cheddar cheese is better than American cheese. P
2. The window is broken. P
3. Turn left at the next light. command, not a sentence
4. Art is a great dancer. P
5. Art is very popular. Q
6. Art is very popular and he is a great dancer. P and Q
7. Art is very popular because he is a great dancer. P
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CHAPTER 6Section 6.1 The Logic of Sentences
8. John and Mary got married. P
9. John and Mary had a baby. Q
10. John and Mary got married and had a baby. P and Q
11. John and Mary had a baby and got married. Q and P
12. Can you hear that music? question, not a sentence
13. Texas is larger than China. P
14. Wanda believes Texas is larger than China. P (have to treat as a single sentence)
You will notice here that sentence 7 is represented by “P,” since, as we saw, we cannot treat its components separately. But given what we saw about “and,” we are able to treat the components of sentence 6 separately and show that structure by representing each com- ponent sentence with individual sentence letters.
You may also want to notice sentences 10 and 11. These sentences help show that logic may treat two sentences differently from how we might treat them in conversation. In ordinary speech, we might think that sentences 10 and 11 are different, each indicating a different order of events—sentence 10 indicating that the marriage took place before the baby was born and sentence 11 indicating that the baby was born before John and Mary got married. But logic doesn’t make this distinction. In logic, the two sen- tences really say the same thing: two things hap- pened, with no indication of which occurred first. We always want to be careful not to introduce more information than the sentences themselves allow us to consider; in these two sentences, the structure does not indicate the sequence of events.
Truth Functions and Bivalence
In the previous section, we looked at truth functions; the basic idea is that we have a truth function if we can determine the truth-value of a longer, compound sentence by knowing the truth-values of the shorter, component sentences that make it up. We used examples of longer sentences with “because” and “believes that” as instances where we can’t determine the truth-value of the longer sentence just by knowing the truth-values of its component sentences. This is the reason that “because” and “believes that” are not truth-functional connectives.
This may sound complicated, but another example may make it clear. Let’s symbolize the sentence “One of George Washington’s little brothers was named John” as P. Sentence P is, in fact, true. But even if we happen to know that this sentence is true, would the following sentence (which we call “Q”) be true?
(Q): It is well-known that one of George Washington’s little brothers was named John.
Stop and Think: Assertoric Sentences
See if you can identify which of the following are assertoric sentences and which are not.
• You must work harder. • Paris is the capital of Hawaii. • Paris is the capital of France.
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CHAPTER 6Section 6.1 The Logic of Sentences
It seems pretty unlikely that Q is true, even if P is true. Q introduces a new issue—how well-known this is—and we can’t be very confident that it is well-known (or not). So here we know the truth- value of the component P but still don’t know the truth-value of the longer sentence Q. Since we can’t move from knowing the truth-value of P to knowing the truth-value of Q, this doesn’t give us a truth function. As we saw earlier with “believes that,” “it is well-known that” also doesn’t work as a truth-functional connective.
We look more closely at these truth-functional con- nectives as we go along. We have already seen that “and” can be such a connective: we can determine the truth-value of P and Q if we know the truth- value of P and we know the truth-value of Q. The connective “or” also can be a truth-functional con- nective: consider the following sentence, which we symbolize as R:
(R): I will take calculus, or I will take accounting.
If we let “P” be “I will take calculus” and “Q” be “I will take accounting,” we know that if P is true and Q is true, R is true. That is, knowing the truth- values of the component sentences—in this case
they are both true—we can determine the truth-value of the larger sentence R. Whatever the truth-value of P is, and whatever the truth-value of Q is, once we know those truth- values, we can determine the truth-value of the larger sentence R. So this gives us a truth function, and “or” operates as a truth-functional connective.
So far in this discussion, we have assumed that there are only two truth-values—that sentences are either true or false. We should probably make this assumption explicit, and to do so we now introduce the term bivalence. Bivalence simply means “two values,” and here we operate on an assumption of bivalence (pronounced buy-VAY-lence). That is, we use only the two truth-values of true and false, and no other truth-value is ever used. This, generally, is a pretty safe assumption, but it may be worth pointing out that there are rea- sons not to assume bivalence. For instance, we may not be willing to say that the following sentence is true or false:
It will rain in Cleveland, Ohio, on October 14, 2056.
Bivalence says that either this sentence is true, or it is not true (false). But given that we do not know now, or have any way of knowing now, whether it is true or false, some logicians and philosophers hesitate to say that this sentence is either true or false. Others also point to sentences that use vague terms, such as “tall” or “bald,” to call bivalence into question. Imagine Steve is six feet, two inches; do we want to say the sentence “Steve is tall” is either
Keith Brofsky/Thinkstock
Bivalence says that a sentence is either true or false, but some logicians hesi- tate, especially if sentences use vague terms like “tall.”
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CHAPTER 6Section 6.1 The Logic of Sentences
true or false? Again, some philosophers and logicians worry about such vague or ambigu- ous terms enough to cause them to hesitate in assuming that all sentences are either true or false, or assume bivalence.
These are complicated—and quite interesting—issues that are mainly studied in the philosophy of logic and the philosophy of language. But in this discussion, we assume bivalence in order to avoid these kinds of complications. Therefore, every sentence we consider has one of two possible truth-values—true or false—and no other truth-values are possible. We saw in Chapter 2 that the kind of logic we are using here treats the past, present, and future versions of sentences the same—technically, we ignore the tense of the sentence—so we do not have to worry about problems with statements about the future. Nor do we have to worry about vague terms; we consider “Frank is bald” to be either true or false; in other words, we treat such sentences bivalently.
Truth-Functional Connectives
We have already seen two truth-functional connectives: “and” and “or.” We saw that we can determine the truth-value of compound sentences that are built with these connec- tives if we know the truth-value of the components of such sentences. We can now make all of this more precise and introduce the remaining standard truth-functional connec- tives. Then we can begin to symbolize sentences and arguments in order to look at their logical structure.
Conjunction (&) Since we already looked at “and” a bit, that is where we start our discussion. As a connec- tive, “and” is symbolized by “&” and is called the conjunction. Let’s take a sentence and translate it into symbols.
Charlotte likes to swim, and Charlotte likes to play basketball.
Let “P” be “Charlotte likes to swim,” and “Q” be “Charlotte likes to play basketball,” which gives us
P and Q
Using the new symbol for conjunction to express “and,” we get
P & Q
We now need to figure out when this conjunction, P & Q, is true. We define such sentences as true in only one situation: when all of its component sentences are true. If it is true that Charlotte likes to swim, and it is true that she likes to play basketball, then it is true that she likes to do both. So “P and Q” is true when P is true and Q is true, and only when P is true and Q is true. If we discover that it is false that Charlotte likes to swim, or false that she likes to play basketball, or false that she likes to do both, then the conjunction as a whole is false. So the conjunction is defined as true only when all of its components are
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CHAPTER 6Section 6.1 The Logic of Sentences
true. We can express this definition in what is known as a truth table, which lists all the possible truth-values and allows us to give a complete display of all sentences that are structured as conjunctions, or as “P & Q.” (We look more closely at truth tables later, to see how they can demonstrate various properties of sentences and arguments.)
P Q P & Q T T T F T F T F F F F F
Truth tables are a handy way of displaying all the possible truth-values that can be assigned to sentences, and they allow us to define the various truth-functional connectives just by using the truth-values of true and false. They can be used to demonstrate the properties of such sentences as those that are necessarily true, or true for all possible truth-values of the component sentences (known as “tautologies”), and those that are necessarily false, or false for all possible truth-values of the component sentences (contradictions). They can also demonstrate other things, including that certain kinds of arguments are valid or not valid. Once you see how they work, truth tables can be extremely useful, although we will have to introduce a bit more material to see all the things truth tables offer.
Disjunction (v) The other connectives are defined in similar ways. Previously, we looked at “or” a bit, and now we can explain it more explicitly by symbolizing it and then giving a truth table that defines it.
The truth-functional connective “or” is written “v”—sometimes called a wedge—and the connective is called the disjunction. So “P v Q” is read as “P or Q.” Using our earlier exam- ple, we now symbolize it and then show its truth function using a truth table.
I will take calculus, or I will take accounting.
Letting “P” be “I will take calculus” and “Q” be “I will take accounting,” we get
P or Q
Using the new symbol for disjunction, we get
P v Q
The disjunction is somewhat different than con- junction, for it is true when either one of its com- ponents is true, or when both its components are true. So this sentence would be true if you take calculus, or if you take accounting, or if you take both calculus and accounting. It is false only when all of its components are false; that is, if
Kablonk! Kablonk!/Photolibrary
With a disjunction connective, you would paint a room peacock blue OR golden wheat, not peacock blue AND golden wheat.
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CHAPTER 6Section 6.1 The Logic of Sentences
you took neither calculus nor accounting, P v Q would be false. This is what its truth table looks like:
P Q P v Q T T T F T T T F T F F F
At first, this may look similar to the previous truth table we saw for conjunction (“&”). But notice the middle two rows; here, with disjunction (“v”), each of these sentences is true because only one of its component sentences is true.
Disjunction, however, brings with it one ambiguity, or complication. Sometimes people use disjunction in a way that means one thing or the other, but not both. Perhaps the server at a restaurant tells you
With your dinner, you get soup or salad.
Generally, we assume this means one gets either soup or salad, but not both. This is what is called “exclusive” disjunction, because it excludes the possibility of both things in the disjunction. This can occasionally cause confusion. Imagine the annoying six-year-old next door is bothering you, as he often does. You tell him, “If you go away for the rest of the day, I will take you to play video games or to get ice cream.” He agrees, and the next day you take him to play video games. A couple of days later, he returns for you to take him for ice cream. He understood “I will take you to play video games or to get ice cream” inclusively—video games, ice cream, or both—while you intended it exclusively—either video games or ice cream, but not both.
Generally, the context in which such disjunctions are used makes it clear whether exclu- sive disjunction or inclusive disjunction was the intended meaning. But we can eliminate any such worries, because we define the “v” symbol to mean inclusive disjunction. So the first row of its truth table, as we saw, indicates that “P v Q” is true when “P” is true and “Q” is true. Whenever we see the truth-functional connective “v,” then we know the sen- tence is true when P is true, Q is true, or both are true. In other words, “v” is inclusive disjunction.
Negation (~) When we negate something, we deny it; for us, that simply means we change the truth- value, from true to false or from false to true. Negation is symbolized using “~” and is often read “not.” The more explicit way to state the negation is “it is not the case that,” but generally “not” does the trick.
Here are a few sentences and their negations:
It is snowing It is not the case that it is snowing I like football I don’t like football She doesn’t watch too much TV It is not the case that she doesn’t watch too much TV
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CHAPTER 6Section 6.1 The Logic of Sentences
As the last sentence indicates, some sentences have negations within them; we negate such a sentence by denying what it says. And for all of these sentences, if we represent the sentence in the left-hand column as “P,” the sentences in the right-hand column are all written as “~ P.” As should be clear, if P is true, then ~ P is false; if P is false, ~ P is true. This gives us a very concise truth table:
P ~ P T F F T
What we also see from this is that two negations—(~ (~ P))—cancel each other out.
In this kind of logic, then, “not-not-P” is the same as “P” (they have the same truth- value). In fact, any even number (2, 4, etc.) of negations cancel each other out. Although it is unusual and can be confusing to see a large number of negations, they are easily dealt with. So, for example, if someone says, “It is not the case that I do not dislike football,” we can represent “I like football” as “P.” This sentence then becomes “~ (~ (~ P))),” we see we have an odd number of negations (here, 3). All but one cancel out, meaning that this sentence is equivalent to “~ P.” This kind of example also shows that sometimes dealing with sentences in their logical structure is easier than dealing with them in natu- ral language.
Conditional (→) Whereas negation is probably the easiest truth-functional connective to understand, the conditional can be a little confusing at first. Let’s first look at some sentences that could come up in conversation:
If it is too hot today, then she will turn on the air conditioning.
If I win the lottery, then I will buy a new car.
If you eat too many chicken wings at night, then you won’t feel very good the next morning.
These are called conditional sentences, for the “if” part of the sentence introduces a condition, and the “then” part of the sentence indicates what fol- lows from that condition. The condition, which is introduced by “if,” is called the antecedent; what follows, indicated by “then,” is called the conse- quent. If we let the antecedent be symbolized by “P,” and the consequent be symbolized by “Q,” then we can represent the three preceding sen- tences by writing them in this way:
P → Q
read as “if P then Q.”
Thinkstock
“If I win the lottery, then I will drive around in a limo and shop all the time” is an example of a conditional statement.
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CHAPTER 6Section 6.1 The Logic of Sentences
Under what circumstances are conditional sentences said to be true (or false)? Philoso- phers have spent a good deal of time investigating this question, for it is not entirely clear, at least at first, what to do when the antecedent of a conditional is false.
Imagine you are in an English class, and your grade is based on three exams. We might describe the situation with this conditional:
If I get A’s on all my exams, I will get an A in the course.
It seems clear that, if in fact you do get A’s on all your exams and you do get an A in the course, we could say that the conditional as a whole is true. It seems equally clear that, if you get A’s on all your exams and you do not get an A in the course, we should say that the conditional states something that is false. But what happens if you do not get A’s on all your exams? In this case, the antecedent is false, and you have not met the condition. Perhaps you do not get A’s on all the exams and still get an A in the course. Or perhaps you do not get A’s on all the exams and you do not get an A in the course. How are these kinds of conditionals treated, then, when the condition (the “if” part of the conditional, or the antecedent) is false?
Various responses have been made to this question; indeed, entire books have been writ- ten about how one handles such a situation. One might think that the conditional as a whole is neither true nor false if the antecedent is false; but in the kind of logic we are considering here, that would be to introduce a third truth-value, which conflicts with our basic assumption of bivalence—that all sentences are either true or false. On the other hand, we might think that the conditional as a whole should be false; but this seems to conflict with common sense. (For more consideration of this particular concern, a nice discussion is offered by the philosopher Peter Suber here: http://www.earlham .edu/~peters/courses/log/mat-imp.htm.) We have three options in giving a truth-value to a conditional statement whose antecedent is false. We can give the conditional a third truth-value, we can give it the value “false,” or we can give it the value “true.” Our assumption of bivalence eliminates the possibility of a third truth-value; more complex reasons than we can consider here eliminate the possibility of giving it the value of false. So we are left with the only other option—that conditionals with false antecedents are given the truth-value of true. This, then, gives us the following truth table, which defines this connective:
P Q P → Q T T T F T T T F F F F T
In other words, a conditional is true except when the “if” part of the sentence (the anteced- ent) is true and the “then” part of the sentence (the conditional) is false.
Here’s another way of thinking of the conditional. Caryn asks her teacher what she needs to do to get an A in her class. The teacher says, “If you get A’s on all the tests, you will get an A in the class.” Caryn asks, “So it couldn’t be true that I get A’s on all the tests and not get an A in the class?” The teacher says that is right.
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CHAPTER 6Section 6.1 The Logic of Sentences
This gives us another way of seeing how to think of the conditional. We can use a truth table to show that since Caryn’s version of the teacher’s statement says the same thing as the “if . . . then” statement, then they will have the same truth functional definition. Let “P” represent “I get A’s on all the tests,” and “Q” represent “I get an A in the class.” Here are our sentences, then, in symbols:
If I get A’s on all my tests, I get an A in class: P → Q
It is not the case that I get A’s on all my tests and not get an A in class: ~ (P & ~ Q)
And here is a truth table showing their truth functions. Since they say the same thing, they have the same truth function:
P Q P → Q ~ P ~ Q (P & ~ Q) ~ (P & ~ Q) T T T F F F T F T T T F F T T F F F T T F F F T T T F T
Biconditional (↔) The last truth-functional connective we look at is called the biconditional. In contrast to the sym- bol for the conditional, the arrow for the bicon- ditional (↔) points in both directions, because it establishes a connection between the component sentences that is stronger than that established by the conditional. The biconditional is read as “if and only if.” We would read “P ↔ Q,” then, as “P if and only if Q.” This establishes the kind of con- nection between component sentences that might be seen in a definition. Here’s an example:
A two-dimensional object with three sides is a triangle if and only if a triangle is a two-dimensional object with three sides [this is just the definition of a triangle]
Here, we can symbolize the first part (“a two-dimensional object with three sides is a triangle”) with P, and symbolize the second part (“a triangle is a two-dimensional object with three sides”) with Q. That gives us:
P ↔ Q
which is read, again, as “P if and only if Q.” Here is another way of stating this: if it is a two-dimensional object with three sides, it is a triangle; and if it is a triangle, it is a two-dimensional object with three sides. As you might have already realized, this is the kind of truth-functional connective one sees with definitions, as one might run across in mathematics.
Stop and Think: Conditional Sentences
Here are some conditional sentences in English. Identify which part is the anteced- ent and which part is the consequent and symbolize them with the truth-functional connective “→.”
• If we buy popcorn at the movies, then we will spend too much money.
• Emma looks tired if she doesn’t get a good night’s sleep.
• If they make us wait any longer, we should go somewhere else.
• If I have to do any more homework, I’m going to scream.
• We should download a different song if they don’t have the one we want.
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CHAPTER 6Section 6.1 The Logic of Sentences
We can return to our earlier example to see the difference between the conditional and the biconditional. Remember our earlier conditional stated
If I get A’s on all my exams, I will get an A in the course.
or, as symbolized
P → Q.
At this point, it is helpful to use the concept of necessary and sufficient conditions, which we discussed in Chapter 3, to explain the conditional, the biconditional, and the difference between them. In this example, P—the antecedent—is the sufficient condition for Q. That is, if P takes place (if you get all A’s on your exams), that is enough to know that Q takes place (that you get an A in the course).
Highlights: Necessary and Sufficient Conditions
In a conditional sentence, such as “if P then Q” (P → Q), the antecedent is said to be the sufficient condition for Q, and Q is the necessary condition for P. These are very helpful terms and can be very useful in understanding conditional sentences. Here are examples of a sufficient condition, a necessary condition, and a necessary and sufficient condition. It is a good idea to get comfortable with this language.
Sufficient Condition If I live in Chicago, then I live in Illinois.
(Here we see that if we know a person lives in Chicago, that is enough to know that that person lives in Illinois; to live in Chicago is a sufficient condition for living in Illinois.)
Necessary Condition If a person is to be elected the president of the United States, he or she must be 35 years old (or older).
Here we see that one must be at least 35 years old to be elected president, although obviously that isn’t enough to get elected. It is then a necessary condition to become president, to be 35 years old or older.
Necessary and Sufficient Condition If Amy is a female parent, then she is a mother; if Amy is a mother, then she is a female parent. (Or Amy is a female parent if and only if she is a mother.)
It is required both, in order to qualify as a mother, to be female and to be a parent; if you are a female and a parent, that’s enough to know you are a mother. So, here, being a female parent is both necessary and sufficient for being a mother (and being a mother is both necessary and sufficient for being a female parent).
A website that goes into more detail on these relationships can be found here:
http://plato.stanford.edu/entries/necessary-sufficient/
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CHAPTER 6Section 6.1 The Logic of Sentences
Let’s imagine the course requirements are much stricter than the ones indicated so far. Perhaps the teacher says the only way to get an A in the course is to get A’s on all the exams. The teacher might put it this way: If you get A’s on all your exams, you will get an A in the course, and if you get an A in the course, you will have gotten A’s on all your exams. Symbolized with our truth-functional connectives, we can write it this way:
(P → Q) & (Q → P)
In other words, P is a sufficient condition for Q, and Q is a sufficient condition for P. We know that if you get A’s on all your exams, you get an A in the course, and it is the only way to get an A in the course. We know one gets an A in the course if and only if one gets A’s on all the exams. This, then, is the biconditional, and as we see, stating the relation- ship with a biconditional can be expressed as we just did: using a conjunction (“&”) of two conditionals, expressing the idea that each antecedent is a sufficient condition for the consequent. Combining the two conditionals, then, gives us the biconditional, which we can define using the following truth table:
P Q P ↔ Q T T T F T F T F F F F T
Perhaps the easiest way to remember this truth-functional connective is by seeing that it is true when the component sentences have the same truth-value. Where both “P” and “Q” are true, “P ↔ Q” is true; where both “P” and “Q” are false, “P ↔ Q” is true. The bicondi- tional is false only when the two sentences have different truth-values.
The Truth-Functional Connectives • “~” not/it is not the case that [negation] • “v” or [disjunction] • “&” and [conjunction] • “→“ if . . . then [conditional] • “↔“ if and only if [biconditional]
P Q ~ P P V Q P &Q P → Q P ↔ Q T T F T T T T F T T T F T F T F F T F F F F F T F F T T
Symbolizing Sentences
Now that we’ve discussed sentence letters—P, Q, and, if we need them, R, S, etc.—and the truth-functional connectives, we can use these to symbolize sentences. On occasion, we also use parentheses to group parts of sentences; this helps make the structure clearer and helps avoid any ambiguity that might otherwise arise.
1. Cynthia can play the guitar and the piano. 2. Either I’m crazy or everyone loves logic.
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CHAPTER 6Section 6.1 The Logic of Sentences
3. If we don’t pay the electric bill, our power will get cut off. 4. Henry likes online education because he can go to class in his pajamas. 5. I will go to the game only if I can get a ticket.
Let’s take each sentence, show how to symbol- ize its components (if any), and then use the truth-functional connectives to give a complete symbolization.
1. Cynthia can play the guitar and the piano.
Let P represent “Cynthia can play the guitar” and Q represent “Cynthia can play the piano.”
P and Q
P & Q
2. Either I’m crazy or everyone loves logic.
Let P represent “I’m crazy” and Q represent “everyone loves logic.”
P or Q
P v Q
Table 6.1 Sentences
In summary, we use two kinds of letters when we symbolize sentences and arguments: sentence letters and truth-functional connectives.
Sentence Letters • Also called variables • The symbols that indicate propositions, sentences • p, q, r, s, t, and so on as needed
Truth-Functional Connectives • The symbols that indicate the relationship between propositions • &, v, ~, →, ↔
Connective Meaning Variables & Connectives Example
& Conjunction (and)
p & q I will take accounting and I will take calculus
v Disjunction (or)
p v q I will take accounting or I will take calculus
~ Negation (not)
~ p I will not take accounting It is not the case that I will take accounting
→ Conditional (if-then)
p → q If I take accounting then I will take calculus
↔ Biconditional (if and only if)
P ↔ q I will take accounting if and only if I take calculus
Thinkstock
Joey goes to school, plays soccer, and plays guitar. How would you symbol- ize that sentence?
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CHAPTER 6Section 6.1 The Logic of Sentences
3. If we don’t pay the electric bill, our power will get cut off.
Let P represent “we pay the electric bill” and Q represent “our power will get cut off.”
If not P then Q
~ P → Q
Here we could have used P to represent “we don’t pay the electric bill,” thus including the negation within the symbolization. We would then symbolize the sentence as “P → Q.”
4. Henry likes online education because he can go to class in his pajamas.
Let P represent the entire sentence; here we cannot break the sentence up into smaller components. As we saw, “because” doesn’t function truth functionally.
5. I will go to the game only if I can get a ticket.
Let P represent “I will go to the game” and Q represent “I can get a ticket.”
P only if Q
One might be tempted to symbolize this sentence “Q → P.” But notice what the sentence says: it says I will go to the game only if I get a ticket. It does not say that if I do get a ticket, I will go to the game. So “only if” here indicates that what follows it is a necessary condition (see the earlier Highlights section on necessary and sufficient conditions); the necessary con- dition then becomes the consequent of the conditional, and so we symbolize this sentence as
P → Q
Practicing symbolizing sentences is a good way to become comfortable with how one goes about making the structure of sentences clear and how truth-functional connectives work. Here is another set of sentences, slightly more complicated:
6. Robyn will be promoted if and only if she is deployed to Iraq. 7. If we go to the beach or the mountains, we will have fun. 8. If I win the lottery, I will not quit my job, but I will buy a new car. 9. Suki likes rice, but her husband does not.
10. They went to the movies, then went to dinner, then went dancing.
Let’s go through these sentences as we did the previous ones, noting some important details as we go along.
6. Robyn will be promoted if and only if she is deployed to Iraq.
Let P represent “Robyn will be promoted” and Q represent “Robyn is deployed to Iraq”
P if and only if Q
P ↔ Q
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CHAPTER 6Section 6.1 The Logic of Sentences
Here we see that “she” in the second part of the sentence obviously refers to Robyn.
7. If we go to the beach or the mountains, we will have fun.
Let P represent “we go to the beach,” Q represent “we go to the mountains,” and R repre- sent “we will have fun.”
If either P or Q, then R
(P v Q) → R
Here, for the first time, we need a third sentence letter, R; the antecedent of this con- ditional is itself a disjunction. We also use parentheses to structure the sentence in the appropriate way.
8. If I win the lottery, I will not quit my job, but I will buy a new car.
Let P represent “I win the lottery,” Q represent “I will quit my job,” and “R” represent “I will buy a new car.”
If P then not Q and R
P → (~ Q & R)
Here we make the negation explicit with the symbols, although we could have just as easily symbolized “I will not quit my job” as P. We also see in this example that “but” is treated as a conjunction, which is the standard way logicians treat “but.” We also use parentheses here to make clear that the two results—relative to the job and the car—are both conditional on winning the lottery.
9. Suki likes rice, but her husband does not.
Let P represent “Suki likes rice” and Q represent “Suki’s husband does not like rice.”
P and Q
P & Q
Here we include the negation in symbolizing “Suki’s husband does not like rice”; we could have made it explicit by using Q to represent “Suki’s husband likes rice,” and sym- bolizing it as ~ Q. We again see that “but” is treated as a conjunction. We also see that the second part of this sentence, “Suki’s husband does not like rice,” is a bit more formal than the way the sentence is actually stated, because “her” clearly refers to Suki and “does not” clearly means “does not like rice.”
10. They went to the movies, then went to dinner, then went dancing.
Let P represent “they went to the movies,” Q represent “they went to dinner,” and R rep- resent “they went dancing.”
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CHAPTER 6Section 6.1 The Logic of Sentences
P and Q and R
P & (Q & R) or equivalently (P & Q) & R
Here we can simply state that the three things occurred; as we saw previously in this chapter, the logic we are using can’t indicate what the sequence or order of events is. Because the only connective used here is the conjunction “&”, it doesn’t matter how we group the sentences using parentheses. If different combinations of truth-functional con- nectives are used, however, you need to be sure to group the sentences in the way indicated by the original sentences. But with more than two sen- tences, as we have here, we need to group them using parentheses. Here, where the connectives are all conjunctions (“&”), how we group them doesn’t matter. But in more complex sentences, and when the connectives are different, that grouping can be very important.
Symbolizing Arguments
Now that we have seen how to represent sen- tences with sentence letters and construct com- plex sentences out of simpler components using truth-functional connectives, we can begin sym- bolizing arguments. Symbolizing arguments has
one big advantage: it allows us to see clearly the structure of the argument, and often that allows us to determine very quickly how good the argument is. Particularly in the case of deductive arguments, focusing on the structure of the argument not only reveals better how the argument works—or fails to work—but can also prevent us from being distracted
by the content of the sentences. Consider the fol- lowing two arguments:
1. If we want effective crime prevention, we should use the death penalty more.
We want effective crime prevention. Therefore, we should use the death penalty more.
2. If we want effective crime prevention, we should abolish the death penalty.
We want effective crime prevention. Therefore, we should abolish the death penalty.
Stop and Think: Symbolizing Sentences
Here are some sentences for you to symbol- ize. As we have done with the previous sen- tences, indicate what you use each sentence letter to represent, and then replace the English connectives (such as “and,” “but,” and “if . . . then”) with our truth-functional logical connectives.
• Mia likes the Dodgers, but Steve likes the Padres.
• If they don’t see that movie, they will be sorry.
• I sent my kids to bed early, and then they were upset.
• If you watch too much television, then you won’t get your work done.
• I will buy you some ice cream only if you wash the car.
• Dan will move to San Francisco if and only if he gets a good job there.
• My boss is helpful, friendly, and generous.
• If we don’t go to church Friday night, then we will go Sunday morning.
• Bob likes bluegrass music, but Carolyn likes jazz and rap.
• The dress didn’t fit, so she took it back and got a refund.
Jupiterimages, Brand X Pictures/Thinkstock
Just like in math, we see that using symbols can help us express a complex equation or argument.
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CHAPTER 6Section 6.1 The Logic of Sentences
If we focus all of our attention on the details of whether the first premise is true, we may get distracted by arguments over the effectiveness of the death penalty. But logic shows us that the first step is to determine whether the argument—here a deductive argument—is valid, and we can do that by symbolizing it with sentence letters and truth-functional connectives:
P → Q
P
Q
Once we determine that the argument is valid, then we turn to the question of whether it is sound. As we saw earlier, a sound argument is a valid deductive argument with prem- ises that are actually true. For arguments 1 and 2, we see that both arguments are valid because of their logical structure. Once we have established that, we can then go on to investigate which premises are true. In this way, we can focus on where the disagreement really lies. But what about this argument?
If prayer is not effective, then I do not get what I pray for.
I do not get what I pray for.
Therefore,
Prayer is not effective.
This may seem to be a rather controversial argument, but the logician need not worry about the effectiveness of prayer at all. The logician can simply point out that the argu- ment is not valid because it commits a fallacy: specifically, the fallacy of affirming the consequent (Q). We can see this most easily by symbolizing the argument (here including the negations within our symbolization):
P → Q
Q
P
We know this argument form is not valid, no matter what the premises say. If the argument is structured in this way, it is never a good argument. We can save a lot of time examining this argument, for we can reject it simply on the basis of its logical structure, which reveals that it commits the fallacy of affirming the consequent.
We can now look at some of the arguments we saw earlier and symbolize them. Perhaps you remember this argument from our earlier discussion:
If we go out to eat tonight, we should go eat pizza.
If we stay in tonight, we should order pizza.
We should either go out for pizza or order pizza.
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CHAPTER 6Section 6.1 The Logic of Sentences
We can symbolize it this way:
P: We go out to eat tonight
Q: We should go eat pizza
R: We stay in tonight
S: We should order pizza
We can then symbolize this argument as follows:
P → Q
R → S
Q v S
This structure makes it fairly easy to see how the argument works. As stated, it seems that it is not valid, until we look at P and R, which state that (P) we go out or (R) we stay in. If we think just a bit, we see that those are probably our only two options, for we are either going to stay home or not stay home. If we interpret “not staying home” as “going out,” then it seems that we have an unstated premise that is pretty easy to accept, namely “We will stay home or go out.” If we make this premise explicit, then the argument is valid, and its logical structure makes that clear:
P → Q
R → S
P v R
Q v S
This argument, known technically as a complex constructive dilemma, really functions in a way similar to the argument form of modus ponens. In modus ponens, we have a condi- tional premise and its antecedent affirmed:
P → Q
P
Q
In the complex constructive dilemma, we simply have two conditional premises, and the antecedent of each is affirmed. Looking at the structure of these logical forms should make it easier to see that both forms fundamentally rely on the same kind of logical inference.
How might this argument be symbolized?
All cats have four legs.
All dogs have four legs.
All cats are dogs.
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CHAPTER 6Section 6.1 The Logic of Sentences
It is probably obvious that this argument is not valid but, again, looking at its structure makes the mistake clearer.
The standard way of treating such claims as “All A are B” is to see that this sentence says, “If something is an A, then that thing is a B.” So saying, “All trout are fish” is to say, “If this is a trout, then it is a fish.”
P: this is a cat
Q: this is a dog
R: this has four legs
Symbolized, then, the argument looks like this:
P → R
Q → R
P → Q
Some arguments—perhaps this one is a good example—don’t require us to symbolize them in order to realize that they are not valid. Others, par- ticularly more complex arguments, may not be so clear. But a fundamental feature of logic is that no matter how complex or confusing the argument, all arguments are constructed and evaluated in the same way. So knowing how to symbolize basic argument forms is very useful in examining those with a more complicated structure.
Let’s imagine a company, called NewBrandToys, that confronts a rather serious business problem. To stay in business, it must make a profit and it
Table 6.2 Advantages and Disadvantages of Truth Tables Advantages
• Assuming they are done correctly, truth tables guarantee results when using them to test for the validity of arguments.
• Truth tables can also determine what kind of truth function a sentence has: whether it is always true (known as a tautology), always false (known as a contradiction), or sometimes true and sometimes false.
• Truth tables can show that sentences are truth-functionally equivalent to each other; sometimes a confusing sentence can be shown to be equivalent to a statement that is much clearer.
• Truth tables display all the possible truth-values involved in a complex sentence or in an argument.
Disadvantages
• Truth tables work only for deductive arguments.
• Truth tables can get very large if there are a lot of different sentence letters (their size increases exponentially).
NASA/Getty Images/Thinkstock
A man on video, with feedback. Tau- tology refers to needless, redundant repetition.
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CHAPTER 6Section 6.1 The Logic of Sentences
must maintain market share. But to make a profit, it must raise its prices; to maintain mar- ket share, it must not raise its prices. Evidently, this company is in trouble. This is what the argument looks like:
If NewBrandToys is to stay in business, then it must raise its prices and not raise its prices.
Therefore,
NewBrandToys is not going to stay in business.
We then assign different sentence letters, and, of course, we don’t have to use P and Q:
N: NewBrandToys is to stay in business
P: NewBrandToys must raise its prices
Assigning truth-functional connectives, we get the final version of the symbolized argument:
N → (P & ~ P)
~ N
We know, of course, that a company cannot both raise its prices and not raise its prices. If it is to stay in business, however, NewBrandToys has to do exactly this. So we know it is not going to stay in business. As we see clearly from the symbolization, for NewBrandToys to stay in business, “P & ~ P” would have to be true. But if P is true, ~ P is false, so this sentence is false, and, of course, if P is false, this sentence is false. “P & ~ P” will always be false—it is a contradiction—so NewBrandToys is in trouble, because the only way it can stay in business is to make a sentence true that will always be false.
Sometimes arguments are rather straightforward, as the ones we have just seen generally are. However, some arguments get much more complicated. It is still good practice to fol- low our procedure in order to see what the argument’s structure looks like. So consider this argument:
If Angela wants to go to college, then she will have to either borrow money or get a scholarship. She gets a scholarship if she does well in high school, but she won’t do well in high school if she hangs out with the wrong friends. If she borrows money, she will have to pay the money back. Angela hangs out with the wrong friends. Angela wants to go to college. So Angela will have to pay back the money.
It may not be obvious whether this argument is valid or not, but it is good practice—and a good first step—to symbolize it. As always, we assign sentence letters to the component sentences and then symbolize the structure of the argument using truth-functional con- nectives. This time we use somewhat different sentence letters, here using letters that refer to the sentences in a natural way, such as “S,” picking up on Angela’s need for a scholar- ship. It is also important to see that the conclusion here is indicated by the word “so.”
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CHAPTER 6Section 6.1 The Logic of Sentences
A: Angela wants to go to college.
B: Angela will have to borrow money.
S: Angela will get a scholarship.
H: Angela does well in high school.
F: Angela hangs out with the wrong friends.
P: Angela will have to pay back the money.
We can now state this argument using these sentence letters:
If A then (B or S)
If H then S
If F then not H
If B then P
F
A
therefore
P
We can now complete the symbolization by apply- ing the appropriate truth-functional connectives:
A → (B v S)
H → S
F → ~ H
B → P
F
A
P
This helps us see the structure of argument bet- ter, but it still may not be obvious whether this argument is valid. Is there a way of determining that the argument here is valid or not valid? For- tunately there is, a method using truth tables. If we symbolize the argument correctly, using the truth table method guarantees that we can find out whether or not this argument is valid. However, in the case of Angela, we also run up against the limitations of truth tables, as we soon see.
Stop and Think: Examining Sentences
Here are a few arguments. Assign sentence letters to the component sentences, and then, using the truth-functional connectives, symbolize the argument.
• Either Mackenzie likes ice cream or Zach likes ice cream. Mackenzie doesn’t like ice cream, so Zack does.
• If you go to the beach, then you will go swimming. If you go swimming, then you will be stung by a jellyfish. If you are stung by a jellyfish, then you will go to the hospital. So if you go to the beach, you will go to the hospital.
• Zelda plays basketball if and only if Agnes plays football. Agnes does not play football; consequently, Zelda does not play basketball.
• If Shanghai is larger than Houston, then Houston is larger than Iowa City. Iowa City is not larger than Houston. Therefore, Houston is not larger than Shanghai.
Here are two sentences and two arguments. Try constructing a truth table for each. What do you discover about the sentences? Are the arguments valid?
Sentences P ↔ ~ P (P & ~ P) → (Q & ~ Q)
Arguments P → (Q v ~ Q) ~ (Q v ~ Q) ~ P ~ (Q & ~ Q) Q v ~ Q
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CHAPTER 6Section 6.1 The Logic of Sentences
6.1 Quiz
1. True or false? Assertoric sentences are sentences that can be assessed in terms of truth and falsehood.
2. True or false? Compound sentences are sentences that contain more than one claim.
3. True or false? Imperatives and questions are two examples of assertoric sentences.
4. True or false? “Rene Descartes was a radical skeptic” is an example of an asser- toric sentence.
5. Which of the following is not an assertoric sentence?
A. Eat your cereal! B. Love is all you need. C. Smoking causes cancer. D. Pets lower blood pressure.
6. What do we mean by the “truth function” of a sentence?
A. That the sentence is, in fact, true. B. That we cannot think of the sentence as true or false. C. That if we know the truth-value of the components of a sentence, we know
the truth-value of a complex sentence made up of each. D. None of the above.
7. True or false? Any assertoric sentence can be represented with a “P” or “Q” variable.
8. What does it mean to say that a sentence is bivalent?
A. That it is a complex sentence made up of two component parts. B. That it has only two possible values: true or false. C. That it contains one false and true subcomponent. D. All of the above.
9. The logical notation for the truth functional connective “and” is
A. v B. ↔ C. ~ D. &
10. The logical notation for the truth functional connective “or” is
A. v B. ↔ C. ~ D. &
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CHAPTER 6Section 6.1 The Logic of Sentences
11. The logical notation for negation is
A. v B. ↔ C. ~ D. &
12. The logical notation for the truth functional connective “if and only if” is
A. v B. ↔ C. ~ D. &
13. True or false? In a sentence containing a conjunction, if at least one of its compo- nents is true, we know the whole sentence is also true.
14. True or false? In a sentence containing a disjunction, if at least one of its compo- nents is true, we know the whole sentence is also true.
15. True or false? When you negate a negated sentence, both negations cancel each other out.
16. True or false? A conditional or “if, then” sentence is represented by the sign “→.”
17. Which of the following is a necessary condition for being over six feet tall?
A. Being less than five feet tall. B. Weighing over 125 pounds. C. Being at least five feet tall. D. Being at least seven feet tall.
18. Which of the following is a sufficient condition for being over six feet tall?
A. Being less than five feet tall. B. Weighing over 125 pounds. C. Being at least five feet tall. D. Being at least seven feet tall.
19. What is the correct symbolization of the following sentence?
Either we go up or we go down. (Let p = “we go up,” q = “we go down”)
A. p → q B. p v q C. p & q D. p therefore q
20. What is the correct symbolization of the following sentence?
The movie was boring, and the movie was far too long. (Let p = “the movie was boring,” q = “the movie was far too long”)
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CHAPTER 6Section 6.2 Truth Tables
A. p → q B. p v q C. p & q D. p therefore q
21. What is the correct symbolization of the following sentence?
If you go to the movie, you will be bored. (Let p = “if you go to the movie,” q = “you will be bored”)
A. p → q B. p v q C. p & q D. p therefore q
Answers 1. True. This type of sentence is the central concern of most systems of formal logic. 2. True. “Cesar ate dessert and Rebecca had a cocktail” is an example of a compound sentence because it
contains two separate claims. 3. False. Neither of these can be thought of in terms of truth or falsehood, so they do not qualify. 4. True. Even though this sentence is in fact false, it qualifies as assertoric because it makes sense to assess
it in terms of “true” and “false.” It is important not to make the mistake that all assertoric sentences must also be true.
5. A. “Eat your cereal” is an imperative and hence has no truth value. 6. C. “Truth function” is a technical term that refers to our ability to deduce truth-value of a whole from the
truth-value of the parts of the whole. 7. True. This is logical shorthand for sentences and enables logicians to more elegantly represent complex
logical statements. 8. B. Rejoinder: Bivalence is attributed to assertoric sentences and refers to the fact that they are either true
or false; there is no other possible value we can attribute to them. 9. D. For example, “P and Q” would be notated as “P & Q.” 10. A. For example, “P or Q” would be notated as “P v Q.” 11. C. For example, “not p” would be notated as “~ p.” 12. B. For example, “P, if and only if Q” would be notated as “P ↔ Q.” 13. False. Conjunctions are true only if both of their components are true. 14. True. So long as one half of an “either/or” statement is true, the entire sentence is true. 15. True. ~ (~ that water is H
2 O means “it is not the case that it is not the case that water is H
2 O,” which in
turn is equivalent to “water is H 2 O.”)
16. True. “If p, then q” translates to “p → q.” 17. C. No one who is over six feet tall is not at least five feet tall as well. Thus, being at least five feet tall is a
necessary, though not sufficient, condition for being at least six feet tall. 18. D. If we know something is over seven feet tall, then we can be assured that it is also over six feet tall.
Thus, being over seven feet tall is a sufficient condition for being over six feet tall. 19. B. The “v” symbol represents “or” in disjunctive sentences. 20. C. The “&” symbol represents “and” in conjunctive sentences. 21. A. The “→” symbol represents the “if, then” relation in conditional sentences.
6.2 Truth Tables
Truth tables are a well-known method for seeing the properties of sentences as well as whether arguments are valid. Here we focus on symbolizing arguments, then testing them for validity using truth tables.
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CHAPTER 6Section 6.2 Truth Tables
Truth Tables and Validity
Truth tables can do a number of things, including testing deductive arguments for validity (which is our focus). They do so by assigning truth-values to the various components of the sentences that make up an argument and then methodically display- ing the truth-values of the sentences that make up the premises and the sentence that serves as the argument’s conclusion. The fundamental idea to remember is our original definition of validity: a deductive argument is valid if accepting the prem- ises as true requires the conclusion to be accepted as true. What we do, then, to test an argument for validity with a truth table is to apply truth-values to determine the truth-value of the premises and to determine the truth-value of the conclusion. If the conclusion is true on every row of the truth table where the premises are also true—if there is no row where the premises are true and the conclusion false—then the argument is valid. This may sound a bit abstract at first; the best way to learn how to test an argument for validity using truth tables is to start testing them!
We can begin with a simple argument:
Dave likes either soccer or hockey.
Dave does not like hockey.
Therefore,
Dave likes soccer.
As usual, we assign sentence letters, and then symbolize the argument:
P: Dave likes soccer
Q: Dave likes hockey
P or Q
not Q
therefore
P
P v Q
~ Q
P
We begin constructing the truth table by listing the sentences that make up the argument (here, just P and Q) in the first columns. We then give a column for each premise, and a column for the conclusion:
P Q P v Q ~ Q P
Daniel Schoenen/Photolibrary
Truth tables help us test an argument for validity.
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CHAPTER 6Section 6.2 Truth Tables
We then apply truth-values to our original sentence letters:
P Q T T F T T F F F
Notice that we give alternating truth-values in the first column, under P, and alternat- ing pairs of truth-values under Q. This guarantees we list all the different possible com- binations of truth-values. Remembering the definitions for the various truth-functional connectives we saw earlier, we then apply those to fill out the remainder of the truth table:
P Q P v Q ~ Q P T T T F T F T T F F T F T T T F F F T F
At this point, using our fundamental idea of validity, we look at the table we have con- structed, focusing only on those rows where both premises (“P v Q” and “~ Q”) are true. Is the conclusion (“P”) true on each one of those rows? We see that there is only one such row on our truth table, the third row. Because the conclusion is true where the premises are true—there is no row where the premises are true and the conclusion is false—then this argument is valid.
Here’s another example:
P → Q
Q
P
We may remember the problem with this argument from our earlier discussion—it is not valid—but now we can show it, using a truth table. (If you forget the truth-functional definition of the conditional [“→“], you can always go back and review it.)
P Q P → Q Q P T T T T T F T T T F T F F F T
Again, we look to see what rows of the truth table show all the premises to be true; in this case, it is the first and second row. But in the second row the premises are both true and the conclusion false, and we need only one row where the premises are true and the conclusion false to show that the argument is not valid. We have thus shown that this argument is not valid (and we may remember that it commits the fallacy of affirming the consequent).
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CHAPTER 6Section 6.2 Truth Tables
Constructing Truth Tables
The best way to learn how truth tables work is to take sentences (and arguments) and construct truth tables for them. Here are the steps one takes to construct a truth table.
1. Identify the sentence letters. 2. Make a column for each sentence letter, and assign truth-values to each. (The
easiest way is to alternate truth-values in the first column—the one farthest to the left.)
3. Identify the component sentences. 4. Make a column for each component sentence. 5. Assign truth-values to the component sentences based on the original truth-
values you assigned to the sentence letters.
Here is a sentence for which one might give a truth table:
[P → (Q v ~ P)] ↔ (~ Q & ~ P)
The complete truth table would have separate columns for these components, from left to right (notice how we go from the smallest parts to the entire sentence in the final column):
P
Q
~ P
~ Q
Q v ~ P
~ Q & ~ P
P → (Q v ~ P)
[P → (Q v ~ P)] ↔ (~ Q & ~ P)
Let’s look at one more argument we saw earlier, that concerning NewBrandToys and the problems it faced: namely, to stay in business, it had to raise and not raise prices. The argu- ment looked like this:
N → (P & ~ P)
~ N
We can then use a truth table to see if, in fact, this is a valid argument.
N P ~ P (P & ~ P) N → (P & ~ P) ~ N T T F F F F F T F F T T T F T F F F F F T F T T
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CHAPTER 6Section 6.2 Truth Tables
Here we see that we have an argument with a single premise (“N → (P & ~ P)”), so we need only check to see where that premise is true. This premise is true on two rows (the second and fourth), and there are no rows where the premise is true and the conclusion false. So this argument is valid.
You may have noticed something in particular about this argument. Looking at the conse- quent of the conditional, “N → (P & ~ P)” is “P & ~ P”, you can see from the fourth row of the truth table, “P & ~ P” is false on all possible truth-values. A sentence that is false under all possible truth-values is called a contradiction; it is a sentence that cannot possibly turn out to be true. This is why NewBrandToys is in trouble, of course: if they want to stay in business, at least according to this argument, they have to do something that cannot be done (raise prices and not raise prices); they have to make a contradiction be true, which it never is. In fact this argument form is very old and very well known—old enough to be referred to by the Latin term for “reduce to absurdity,” reductio ad absurdum. In this case, it is absurd to raise and not raise prices; showing that it must do so to stay in business is to show that the hope of staying in business under such conditions makes no sense, or is absurd.
Once you see how they work, truth tables are generally easy to use and have one signifi- cant advantage: they are guaranteed to tell you whether an argument is valid or not (as long as they are set up correctly). But it was mentioned earlier that there are certain limita- tions to truth tables, and the argument we saw with Angela on page 158 shows this clearly.
Table 6.3 Common Deductive Argument Forms Common Deductive Argument Forms (valid) Modus Ponens Modus Tollens
If p then q If p then q
p Not q
Therefore q Therefore not p
Hypothetical Syllogism Hypothetical Syllogism
If p then q p and q
If q then r not p
Therefore if p then r therefore q
Formal Fallacies (invalid) Denying the Antecedent Affirming the Consequent
If p then q If p then q
Not p q
Therefore not q Therefore p
We saw with the truth-functional connection of negation (“~”) that the truth table had only two rows, for we needed only one sentence letter to display all the possibilities (for a sen- tence is either true or false). When we used two sentence letters to show the definitions of the other connectives, the truth table had four rows. In short, if an argument has one sentence letter in it, it requires two rows; if it has two sentence letters, it requires four rows. To display all the possible different truth-values for three sentence letters, we would need eight rows. You probably see the pattern (two, four, eight); truth tables grow very large, very fast. (If you’re curious, the size of a truth table is determined by taking the number two, and raising
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CHAPTER 6Section 6.2 Truth Tables
it to the power of however many sentence letters make up the argument. So two raised to the power of two is four, to the power of three is eight, to the power of four is sixteen.)
The argument we saw about Angela has six separate sentence letters; that would require a truth table with sixty-four rows (26 = 64) to give all the possible combinations of truth- values. That’s an awfully big truth table, and unless you’re a computer or have a great deal of spare time on your hands, you might want to look for better ways to show argu- ments are valid (or not valid). Thus we see that truth tables have their limitations, but if you want to make absolutely sure whether an argument is valid, you can always use a truth table as we have just done, and—assuming you set it up correctly—you are guar- anteed to get your question answered.
6.2 Quiz
1. True or false? Truth tables are unable to determine whether or not an argument is valid.
2. True or false? In a completed truth table, if there is no row in which all the prem- ises are true and the conclusion false, the argument is valid.
3. Which of the following is not a step in the construction of a truth table?
A. Determine whether all the premises are actually true. B. Label the sentences with letters. C. Make a column for each letter and assign a truth-value (T and F) to each. D. Identify the component sentences.
4. True or false? Truth tables may be used on inductive as well as deductive arguments.
5. True or false? While truth tables will work on complex arguments, they can become quite long and time-consuming.
Answers 1. False. Determining validity is the main function of truth tables. 2. True. Truth tables “look at” all possible combinations of truth-values for the premises and conclusion to
determine if in any instance all the premises can be true and the conclusion false. If not, then the argument is valid.
3. A. Remembering that logic deals with “the big if,” we do not concern ourselves with the actual truth- values of the sentences when making truth tables.
4. False. Truth tables are designed to determine validity, and inductive arguments are not assessed in terms of “valid” and “invalid.”
5. True. As long as the argument contains assertoric sentences using standard truth functional connectives, a truth table can be used to determine validity. However, when there are a large number of sentence letters, they can be extremely large and cumbersome.
What Did We Find?
• We saw how symbols can be applied to sentences and then to arguments. • We looked at the notion of a truth function and how some sentences can be
treated truth functionally, and how others cannot be.
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CHAPTER 6Some Final Questions
• We examined how truth tables can be constructed to evaluate certain kinds of sentences as well as to test deductive arguments for validity.
• We used basic symbolic logic to examine some earlier material and see how it can be made more precise.
• We used truth tables to look more formally at arguments we had examined informally.
• We saw that we can construct conditionals that correspond to arguments and that the conditionals that correspond to valid arguments are always true, or are tautologies.
Some Final Questions
• What are the advantages of using symbols to represent sentences and arguments? • How can we use truth tables to determine whether arguments are valid or not
valid? What other properties of sentences can we use truth tables to establish? • Why can’t truth tables show that an argument is sound? • Why can’t we use truth tables to evaluate inductive arguments?
Web Links
Paradoxes of Material Implication
For more consideration of truth tables, a nice discussion is offered by the philosopher Peter Suber here:
http://www.earlham.edu/~peters/courses/log/mat-imp.htm
Necessary and Sufficient Conditions
See the Stanford Encyclopedia of Philosophy for more on necessary and sufficient conditions:
http://plato.stanford.edu/entries/necessary-sufficient/
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